110 HISTORY OP THE THEORY OF NUMBERS. [CHA*. iv H. F. Baker22 extended Sylvester's theorem to any modulus N: ,,(*>_! *w\tf>mi} , , An - ss S -i- - i- (mod A"), A t-i Nmt where the m,- denote the integers <N and prime to JV, N'N=l (mod r), and |&[ is the least positive residue modulo r of k. Lerch23 extended MirimanofTs12 formula to the case of a composite modulus ra. Set g(a,m)=i(a*(m)-l). Let a belong to the exponent <j>(m)/e. Then q(at w)=eSa//3 (mod m), where ft ranges over the residues of the incongruent powers of a, and ma-H3=0 (mod a), 0^a<a. As an extension of Sylvester's theorem, q(a, m) ssS- = 2J (mod m), where v ranges over the integers < m and prune to m, while 7wv+j>==0, mrS-vz=0 (mod a), 0^r,<a, Ogr/<a. For m=mi. . .mk) where the m,- are relatively prune, k q(a} m) s 2 %%'<#>(H;-)$(a, w.,-) (mod m), where m mjn,j, n/2n/=l (mod %). H. Hertzer24 verified that, for a<p<307, a1""1-! is divisible by p3 only for a = 68, p = 113; a = 3, 9, p = ll. He examined all the primes between 307 and 751, but only for a and p a when a< Vp^ finding only p = 113, a = 68. Removing the restriction a< \/pl he found only the solutions p = ll,a = 3; p = 331,a = 18,71; p = 353,a = 14; p= 487, a = 10, 175; p- 673, a = 22, together with the square of each a. A. Friedmann and J. Tamarkine25 gave formulas connecting gu with Bernoullian numbers and [u/p]. A. Wieferich26 proved that if xp+yp+zp = Q is satisfied by integers Xj y, z prime to p, where p is an odd prime, then 2P~1=U (mod p2). Shorter proofs were given by D. Mirimanoff27 and G. Frobenius.28 D. A. Grave29 gave the residue of q% for each prime p< 1000 and thought he could prove that 2P 2 is never divisible by p2 (error, Meissner33). A. Cunningham30 verified that 2P 2 is not divisible by p2 for any prime p < 1000, and31 that 3P - 3 is not divisible by p2 for a prime p = 2a36 H- 1< 100. W. H. L. Janssen van Raay32 noted that 2P 2 is not divisible by p2 in general. wProc. London Math. Soc., (2), 4, 1906, 131-5. 23Comptes Rendus Paris, 142, 1906, 35-38. "Archiv Math. Phys., (3), 13, 1908, 107. MJour. fiir Math., 135, 1909, 146-156. J6Jour. fur Math., 136, 1909, 293-302. 27L'enseignement math., 11, 1909, 455-9. 28Sitzungsber. Ak. Wiss. Berlin, 1909, 1222-4; reprinted in Jour, fiir Math., 137, 1910, 314. 2tAn elementary text on the theory of numbers (in Russian), Kiev, 1909, p. 315; Kiev Izv. Univ., 1909, Nos. 2-10. "Report British Assoc. for 1910, 530. L'interme'diaire des math., 18, 1911, 47; 19, 1912, 159. Proc. London Math. Soc., (2), 8, 1910, xiii.